EFFECTIVE STRESS CHANGE AND POST-EARTHQUAKE SETTLEMENT PROPERTIES OF GRANULAR MATERIALS SUBJECTED TO MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR

SOILS AND FOUNDATIONS Vol. 51, No. 5, 873 884, Oct. 2011 Japanese Geotechnical Society EFFECTIVE STRESS CHANGE AND POST-EARTHQUAKE SETTLEMENT PROPERTIES OF GRANULAR MATERIALS SUBJECTED TO MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR HIROSHI MATSUDA i),andre PRIMANTYO HENDRAWAN ii), RYOHEI ISHIKURA iii) and SATOSHI KAWAHARA iv) ABSTRACT In order to investigate the ešect of cyclic shear direction on the properties of saturated granular materials, such as ešective vertical stress reduction and post-earthquake settlement, several series of multi-directional cyclic simple shear tests under constant volume conditions are performed on Toyoura sand and granulated blast furnace slag (GBFS), as an alternative material. The GBFS has particular properties such as light weight, high shear strength and high permeability, and it is considered to be one of the most promising materials in geotechnical engineering. From the test results, it is clariˆed that the shear strain amplitude has a signiˆcant ešect on the changes in the ešective stress of granular materials. However, at higher levels of shear strain amplitude, the cyclic shear direction has little in uence on the ešective stress reduction. It is found that the vertical strain, after the cyclic shearing of the GBFS samples, was lower than that of the Toyoura sand under the same test conditions. Finally, to evaluate the changes in ešective stress under uni-directional and multi-directional cyclic simple shear conditions, an estimation method is represented by a function of cumulative shear strain G* and resultant shear strain G. The validity of this proposed model is conˆrmed by comparing the experimental and the calculated data obtained under multi-directional cyclic simple shear conditions. Key words: cyclic shear, ešective stress, granular material, liquefaction, settlement (IGC: D6/D7) INTRODUCTION ManystudiesonthechangesineŠectivestressduring an earthquake and the post-earthquake settlement of the ground have been carried out (Lee and Albaisa, 1974; Ishihara and Okada, 1982; Tatsuoka et al., 1984; Tokimatsu and Seed, 1987). Mostly, however, they focus only on cases of uni-directional cyclic shear. In fact, it is widely known that the shear strain during earthquakes shows multi-directional hysteresis paths. Figure 1 presents the orbit of shear strain obtained from the time history of acceleration for the Hyogo-ken Nambu Earthquake of January 17, 1995, for which a multi-directional strain history of the ground can be seen. Several experiments have also been performed to investigate the changes in ešective stress and the settlement induced by multi-directional shaking. Pyke et al. (1975) have shown that the settlement induced by one-directional shaking on a sand layer was less than the settlement measured during multi-directional shaking. Ishihara and Yamazaki (1980), and Tokimatsu and Yoshimi (1982), Fig. 1. Orbit of shear strain in the ground at Hyogo-ken Nanbu Earthquake, January 17, 1995 (Matsuda et al., 2004) i) ii) iii) iv) Professor, Graduate School of Science and Engineering, Yamaguchi University, Japan (hmatsuda@yamaguchi-u.ac.jp). Doctoral Student, ditto; Lecturer of Brawijaya University, Indonesia. Assistant Professor, Graduate School of Science and Engineering, Yamaguchi University, Japan. Master Student, ditto. The manuscript for this paper was received for review on September 29, 2010; approved on May 20, 2011. Written discussions on this paper should be submitted before May 1, 2012 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month. 873

874 MATSUDA ET AL. clariˆed that the liquefaction resistance under multidirectional shaking became smaller than that under unidirectional loading. Nagase and Ishihara (1988) investigated the volume change characteristics of sand induced by undrained cyclic simple shear tests and concluded that the volumetric strain after liquefaction is in uenced by the maximum shear strain amplitude. Ishihara and Yoshimine (1992) proposed an alternative method to estimate the actual liquefaction-induced settlement of the ground. Studies on the settlement of a saturated clay layer in cyclic simple shear tests have also been performed by Ohara and Matsuda (1988), and Matsuda (1997) proposed a simple method to predict the accumulation of excess pore water pressure in clay layers during earthquakes. Furthermore, Matsuda and Nagira (2000) found the existence of a hyperbolic relationship between the ešective stress reduction and the shear strain amplitude of clay layers subjected to uni-directional cyclic shear under undrained conditions. In addition, based on multi-directional drained simple shear tests on Toyoura sand, Fukutake and Matsuoka (1989) proposed the use of parameters for the cumulative shear strain and the resultant shear strain in order to generate a uniˆed law for dilatancy. They found that there is a speciˆc relationship between the cumulative shear strain and the volumetric strain, which is associated with the cyclic dilatancy property of granular materials. Later, Matsuda et al. (2004) and Tamada et al. (2008) performed multi-directional cyclic shear tests under constant volume conditions to investigate the ešect of the cyclic direction on the cyclic properties of saturated granular materials. Moreover, as an alternative material, which is more crushable, granulated blast furnace slag (hereinafter referred to as GBFS) was used in this study. GBFS is considered to be one of the most promising materials due to its geotechnical properties, namely, light weight, high shear strength and high permeability. Compared to natural sand, GBFS has a lot of air bubbles inside its particles; therefore, the particles can be crushed easily by the applied load (Matsuda et al., 2006). Applications of GBFS to an earthquake-resistant earth structure and a light-weight embankment in Japan have also been investigated (Matsuda et al., 2003, 2008). Owing to the fact that its application is on the rise, it has become necessary to study the characteristics of the ešective stress reduction and the post-earthquake settlement of GBFS induced by cyclic loading. Based on these reviews, it seems that studies on the ešect of cyclic shear direction on granular material properties are still limited. The main objective of this paper, therefore, is to investigate the ešect of the cyclic shear direction on the properties of saturated granular materials, such as ešective vertical stress reduction and postearthquake settlement, during multi-directional cyclic simple shear tests. Several series of tests are performed on Toyoura sand and GBFS under constant volume conditions. Shear strain parameters (cumulative shear strain and resultant shear strain) are used to evaluate the relationship between post-earthquake settlement (vertical strain) and the ešective stress reduction of granular materials. Then, an estimation method for the changes in ešective stress is also proposed. The validity of this method is conˆrmed by comparing the experimental and the calculated data obtained under multi-directional cyclic simple shear conditions. MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR TESTS UNDER CONSTANT VOLUME CONDITIONS Basic Properties of Apparatus Figure 2 shows an outline of the multi-directional cyclic simple shear test apparatus. The Kjellman type of shear box is used here. This apparatus can provide any type of cyclic displacement to the bottom of a specimen from two perpendicular directions by the electro-hydraulic servo system. A predetermined vertical stress can be applied to the specimen by an electric-controlled aeroservo system. The specimen is covered with a rubber membrane and is placed in contact with the inner side of 10 acrylic rings, which are stacked. Each acrylic ring is 2.0 mm in thickness. By this arrangement, the specimen is 75 mm in diameter and 20 mm in height. Lateral swelling of the specimen is prevented, while shear deformation is permitted. This test apparatus is shown in Photo 1. This apparatus is the same as that used previously to investigate the undrained response of soils subjected to multi-directional cyclic simple shear conditions, conducted by Matsuda (1997), Matsuda and Nagira (2000) and Matsuda et al. (2004). Photo 2(a) presents the situation of a soil specimen in the shear box, while Photo 2(b) shows the ˆnal setting of the shear box. During cyclic loading, the acrylic rings slide smoothly against each other, and thus, constant volume conditions can be achieved. To minimize friction between the acrylic rings, dry talcum powder (hydrated magnesium silicate) was applied on the surface of each ring. It helps by ˆlling in the microscopic roughness on the rings' surface so that friction can be minimized. By using the same type of shear box, Ohara and Fig. 2. Outline of multi-directional cyclic simple shear test apparatus

MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR 875 Matsuda (1988) performed two-way strain-controlled cyclic simple shear tests under undrained conditions to evaluate the relationship between the excess pore water pressure and the post-earthquake settlement of a saturated clay layer. It is seen that the test results were not in- uenced by any friction from the rings. Photo 1. Multi-directional cyclic simple shear apparatus Samples and Methods The materials used in this study were Toyoura sand and GBFS. The particle size distribution curves and the physical properties of these materials are shown in Fig. 3 and Table 1, respectively. The maximum and minimum dry densities were obtained via the Japanese standard method (JGS,2000).ItisseenthattheGBFShasalmostthesame unit weight of soil grains as the natural sand. Photo 3(a) presents the GBFS material. GBFS is mainly used to produce blast furnace cement in Japan. As an alternative material, GBFS is considered to be one of the most promising materials in geotechnical engineering, because it has particular properties such as light weight, high shear strength and high permeability (Matsuda et al., 2003). From the SEM photograph, it can be observed that GBFS particles are porous on the surface, but have closed pores inside of them (Photo 3(b)). These pores are formed when the melted blast furnace slag is quickly cooled down by pressurized water (Matsuda et al., 2000). Compared with natural sand, therefore, GBFS particles are easily crushed by the applied load. Under multi-directional cyclic shear, the amount of particle crush and settlement are larger than those induced by uni-directional cyclic shear (Matsuda et al., 2006). The performance of GBFS as an earthquake-resistant material and in light-weight embankments has also been investigated. It has been clariˆed that its shear strength increases with time due to its hydraulic properties under ordinary Fig. 3. Grain size distribution of samples Table 1. Physical properties of sample Toyoura sand GBFS Photo 2. Conˆguration of shear box: (a) Soil specimen in shear box, (b) Final setting of shear box Density of soil particles r s (g/cm 3 ) 2.637 2.643 Maximum void ratio e max 0.991 1.510 Minimum void ratio e min 0.630 1.033

876 MATSUDA ET AL. Table 2. Materials and test conditions Materials D r (z) Period T (s) Strain amplitude g (z) Phase dišerence u (9) 0.1 0, 20, 45, 70, 90 70 2 0.3 0, 20, 45, 70, 90 1.0 0, 20, 45, 70, 90 Toyoura sand 2.0 0, 20, 45, 70, 90 0.1 0, 45, 90 90 2 1.0 0, 45, 90 2.0 0, 45, 90 0.1 0, 45, 90 GBFS 70 2 0.3 0, 45, 90 1.0 0, 45, 90 2.0 0, 45, 90 Photo 3. Photograph of GBFS materials: (a) Visualization of GBFS materials, (b) SEM photograph of GBFS particle (Matsuda et al, 2006) natural wet conditions (Matsuda et al., 2003; Matsuda et al., 2008). A report on using GBFS, as a useful material, has already been published (JGS, 2010). Details on the test conditions are shown in Table 2. The main focus of this test program was to investigate the ešect of cyclic direction (phase dišerence) on the undrained response of Toyoura sand under multi-directional cyclic simple shear conditions. As the main material, specimens of Toyoura sand were tested under two densities, namely, (a) medium-dense conditions at a relative density of 70z and (b) higher density conditions at a relative density of 90z. A limited number of multi-directional cyclic simple shear tests was also carried out on medium-dense GBFS material (D r =70z) in order to clarify the dišerence in undrained cyclic responses between GBFS and Toyoura sand. Each specimen was prepared as follows: saturated granular materials were poured into the shear box at a predetermined relative density D r, and the specimen was consolidated for 15 minutes under the vertical stress of s? v =49 kpa. During cyclic shear, the vertical displacement was restricted to keep the volume of the specimen constant. After cyclic shear, the specimen was re-consolidated for 15 minutes under the same vertical stress, and then the drainage valve was opened to drain out the water from the specimen. Finally, the vertical settlement was measured to infer the post-earthquake settlement. Typical shear waves during the multi-directional cyclic Fig. 4. Cyclic shear waves simple shear tests are shown in Fig. 4. Figure 4(a) shows the shear wave form during the uni-directional cyclic sheartests,inwhichshearstrainwasappliedtothespecimen in only one direction (the X direction in this case). Figure 4(b) shows the shear wave during the multi-directional cyclic shear tests, in which shear strain was applied simultaneously in both X and Y directions. The Y axis intersects perpendicularly to the X axis at the center of the bottom part of the specimen. Figure 5 shows the typical deformation of a specimen, conceptually. The shear strain amplitude, g, canbeobtained as a ratio of horizontal displacement d and the initial height of the specimen. For each test, a predetermined phase dišerence u for waves in X and Y directions can be applied. When phase dišerence u is 09, itshowsa uni-directional cyclic shear condition. For the case of the multi-directional cyclic shear tests, when phase dišerence u is 909(or p/2 in radian), it is commonly known as the gyratory cyclic shear condition. The typical patterns of shear strain for specimens under various phase dišerences are shown in Fig. 6. It can

MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR 877 Fig. 6. Fig. 5. Typical deformations of specimen Patterns for shear strain under various phase dišerences be observed from the ˆgure that the phase dišerence ašects the patterns of shear strain. This dišerence is assumed to be an important factor in determining the cyclic deformation properties of soil. Strain-controlled Undrained Cyclic Shear Tests Liquefaction problems in granular soils have commonly been investigated in soil mechanics laboratories using the stress-controlled approach. In this approach, as the number of cycles increases, the cyclic strain increases, eventually becoming a signiˆcant increase. In many tests on loose sand, after the initial liquefaction has been attained, the shear resistance of the specimen decreases signiˆcantly and large deformations develop in the specimen, reaching uncontrolled values. Under such conditions, the experiment should be stopped, and consequently, the results do not completely cover the process after the initial liquefaction (Talaganov, 1992). As an alternative to stress-controlled tests, Dobry et al. (1982) introduced the strain-controlled approach and clariˆed that shear strain is the main parameter that controls the settlement and the excess pore water pressure generation of sand during cyclic loading. In cyclic simple shear tests, the liquefaction parameters are expressed in terms of shear strain amplitude g and the number of cycles n; these main components govern the deformation characteristics of the soil skeleton during cyclic loading (Vucetic, 1992). During cyclic strain-controlled tests, as the number of cycles increases, the pore water pressure continuously increases, and at the same time, the initial shear resistance decreases to smaller values. The failure of a specimen was usually deˆned as the state at which the initial vertical ešective stress had become nearly equal to zero. This approach is advantageous in that experiments can normally be continued until part of the post-liquefaction, and Talaganov (1996) indicates that this approach is useful for characterizing the complete liquefaction process. In order to investigate the properties of saturated granular materials, such as ešective stress reduction and post-earthquake settlement, all of the experiments in this study were performed under conditions of strain-controlled multi-directional cyclic simple shear. The shear strain amplitude was set in the range of g=0.1z to 2.0z. The same values for the shear strain amplitude in the X-andY-directions, (g X)and(g Y),wereappliedtothe specimen throughout the tests. The number of cycles, n, was set in the range from n=1 to150.thewaveform of the applied shear strain is sinusoidal with period T=2.0 s. Figure 7 shows the results of multi-directional cyclic simple shear tests performed on Toyoura sand (D r = 70z). A soil specimen is subjected to a constant level of shear strain amplitude g=±1.0z with an initial vertical stress of 49 kpa. For the case of multi-directional cyclic shear tests with u=909, shear strain is applied simultaneously in both X and Y directions (Fig. 7(a)). Under these undrained cyclic loading conditions, excess pore water pressure developed in the soil specimen and reduced its ešective stress. As a result, initial ešective vertical stress s? vo decreased as the number of cycles increased (Fig. 7(b)). When the ešective vertical stress becomes zero, it indicates that the soil specimen has lique- ˆed. This study mainly focuses on the reduction of ešective stress induced in the specimens, as shown in this ˆgure; therefore, it is not necessary to measure the shear stress during the tests. Deˆnition of Strain Path Parameters Based on the dilatancy concept, Fukutake and Matsuoka (1989) proposed a model to simulate the particle movement of granular materials under drained multidirectional cyclic shear conditions. In this model, the shear strain path on the horizontal plane can be represented as resultant shear strain G, which shows the radial distance from the origin, and cumulative shear strain G*, which shows the length along the shear strain path. Cumulative shear strain G* and resultant shear strain G are expressed as follows:

878 MATSUDA ET AL. Fig. 8. EŠective stress change versus number of cycles for g=0.1% Fig. 7. Typical results from a strain-controlled undrained multi-directional cyclic simple shear test G*=SDG*=S Dg 2 X+Dg 2 Y (1) G= g 2 X+g 2 Y (2) Here, Dg X and Dg Y denote the shear strain increments in the X and Y directions on the horizontal plane, respectively. In the present study, the beneˆts of these parameters were adopted to investigate the correlation between post-earthquake settlement and changes in the ešective stress of granular materials under undrained cyclic shear conditions. TEST RESULTS AND DISCUSSION Changes in EŠective Vertical Stress during Multi-directional Shearing for Various Phase DiŠerences Typical records of the changes in ešective vertical stress in Toyoura sand during multi-directional shearing under various phase dišerences are shown in Fig. 8 (for g=0.1z) andfig.9(forg=1.0z). In general, the ešective vertical stress decreases as the number of strain cycles increases. When the ešective vertical stress reaches zero, it signiˆes that there has been a total loss of the shear strength of the soil or that the liquefaction condition has been reached. It can be observed that the reduction in ešective vertical stress in the case of multi-directional shear is larger than that in the case of uni-directional shear (u=09). From Fig. 9, it is apparent that a higher level of shear strain amplitude implicates the sudden Fig. 9. EŠective stress change versus number of cycles for g=1.0% decrease in ešective vertical stress. It is shown that all of the specimens reached liquefaction in a very short period of time. Mostly, the number of cycles required to reach liquefaction was less than 5, regardless of the direction of cyclic shear. These results show that shear strain amplitude has a signiˆcant ešect on the liquefaction resistance of granular materials. As a result of the undrained cyclic shear tests under constant volume conditions, initial ešective vertical stress s? vo decreases due to the cyclic shear strain. It may be assumed that an increase in excess pore water pressure is taken to be equal to the decrease in ešective vertical stress, namely, u dyn =-Ds? v. In addition, in order to show the beneˆcial use of the cumulative shear strain parameter, the decrease in ešective vertical stress during cyclic shear is presented in Figs. 10 and 11. In these ˆgures, the ešective stress reduction ratio, deˆned by `Ds? v/s? vo`, is plotted against the cumulative shear strain for various phase dišerences, in which Ds? v represents the decrease in ešective vertical stress and s? vo denotes the initial ešective vertical stress. In this series of experiments, the number of cycles n was set to reach n=150. The results show that the ešective stress reduction ratio increases as a function of cumulative shear strain at each phase dišerence. However, it is seen in Fig. 11 that the

MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR 879 Fig. 10. EŠective stress reduction ratio versus cumulative shear strain for g=0.1% Fig. 12. Relationships between the number of cycles required for liquefaction and shear strain amplitude Fig. 11. EŠective stress reduction ratio versus cumulative shear strain for g=0.3% Fig. 13. Relationships between the cumulative shear strain required for liquefaction and shear strain amplitude dependency of `Ds? v/s? vo` on G* is negligible for all the various phase dišerences. It is apparent that with a higher level of shear strain amplitude, the phase dišerence has little in uence on the ešective stress reduction ratio. It has been clariˆed through many experiments that granular soils under multi-directional shearing liquefy easily following the application of cyclic loading, for example, as stated by Ishihara and Yamazaki (1980) and Tokimatsu and Yoshimi (1982). Figure 12 shows the relationship between the number of cycles required for liquefaction and the shear strain amplitude under various phase dišerences for Toyoura sand. For the same shear strain amplitude, as the phase dišerence increases, the number of cycles required to reach liquefaction decreases. Thus, it is recognized that multi-directional shearing reduces the liquefaction resistance of granular soils. Moreover, it is seen that when specimens are subjected to strain amplitude higher than 2z, a very few number of cycles can initiate liquefaction. Using a conservative method, it is di cult to obtain the number of cycles required to reach liquefaction under these conditions. On the other hand, an evaluation of liquefaction using the cumulative shear strain is advantageous in that even after a very few number of cycles, the cumulative shear strain can still be measured. Figures 10 and 11 present the measurements of the cumulative shear strain during the tests. It is observed that the values for the cumulative shear strain can be clearly measured until the occurrence of liquefaction. Thus, it is more accurate to apply the cumulative shear strain to obtain the liquefaction condition in a very short duration. In addition, it may be useful to plot the cumulative shear strain and the shear strain amplitude, as shown in Fig. 13. This ˆgure shows the relationship between the cumulative shear strain and the shear strain amplitude at dišerent cyclic directions for Toyoura sand. For the same shear strain amplitude, it is observed that the cumulative shear strain decreases if the phase dišerence increases. From this ˆgure, it is seen that the curve of cumulative shear strain tends to converge to a value of G* of about 20z. This value of G* corresponds to the minimum cumulative shear strain required to reach liquefaction. Relationship between Cumulative Shear Strain and Postearthquake Settlement In laboratory tests, the post-earthquake settlement was interpreted as changes in volume of the saturated soil specimens caused by the dissipation of excess pore water

880 MATSUDA ET AL. Fig. 14. The variation of vertical strain with cumulative shear strain (Toyoura sand g=0.1%) Fig. 16. Relationships between the vertical strain and cumulative shear strain with dišerent shear strain amplitude (Toyoura sand) Fig. 15. The variation of vertical strain with cumulative shear strain (Toyoura sand g=1.0%) Fig. 17. Relationships between the vertical strain and cumulative shear strain with dišerent shear strain amplitude (GBFS) pressure following the undrained cyclic shear tests. In the present study, the term ``vertical strain'' (e z ) refers to the post-earthquake settlement, which is deˆned as the ratio between the settlement that occurred and the initial height of the soil specimen. The ešects of the phase dišerence on the relationship between vertical strain and cumulative shear strain for ToyourasandareshowninFigs.14and15,forg=0.1z and g=1.0z, respectively. As shown in these ˆgures, the vertical strain (or the post-earthquake settlement) increases with an increasing cumulative shear strain. It is also found that the in uence of the phase dišerence on vertical strain is small under the same cumulative shear strain regardless of the shear strain amplitude. Figure 16 shows the relationship between vertical strain and cumulative shear strain for Toyoura sand under dišerent levels of shear strain amplitude. As shown in this ˆgure, higher levels of shear strain amplitude show larger amounts of vertical strain under the same cumulative shear strain. Figure 17 also presents the relationship between vertical strain and cumulative shear strain for GBFS, which shows a similar tendency to that for Toyoura sand. From these results, it is seen that cyclic shear strain amplitude plays an important role in the settlement characteristics of granular materials. However, it is clari- ˆed that the vertical strain in the GBFS samples is lower than that in the Toyoura sand. This may be due to the particle shape or the roughness properties of GBFS, in which such a material will be ``locked up'' by cyclic strain. EŠect of Cyclic Shear Direction on Cyclic Movement of Particles Figure 18 shows the path of the ešective stress in relation to the shear strain under uni-directional cyclic shearing (u=09) for Toyoura sand. As a strain-controlled test, the shear strain amplitude was set at ±1.0z. The ešective stress decreases rapidly at the beginning of the cyclic shearing, and the rate of decrease slows down when the ešective stress reduction ratio approaches unity. In this case, the loading-unloading paths vary with the cyclic shear strain. This is due to the decrease in the shear resistance of the granular materials. It is apparent that during cyclic shearing, the rate of the changes in ešective stress depends on the level of ešective stress. Further, it is more important to plot the variation in

MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR 881 Fig. 18. Change of ešective stress reduction ratio under uni-directional cyclic shear Fig. 20. Relationships between the cumulative shear strain and ešective stress reduction ratio under multi-directional cyclic shear on Toyoura sand (u=909) Fig. 19), the curves uctuate. In the case of multi-directional shear, however, the ešective stress decreases smoothly, as shown in Fig. 20. This may be due to the dišerence in the cyclic movement of the particles. Matsuda et al. (2004) showed that in the case of uni-directional shear, each particle moves two-dimensionally over the other particles. On the other hand, in the case of multidirectional shear, each particle moves two- or threedimensionally keeping their relative positions in the other particles. Fig. 19. Relationships between the cumulative shear strain and ešective stress reduction ratio under uni-directional cyclic shear on Toyoura sand ešective stress reduction ratios with the cumulative shear strain parameter, as shown in Figs. 19 and 20. These ˆgures present the ešect of the cyclic shear direction on the cyclic movement of the particles. Figure 19 shows the relationship between the cumulative shear strain and the ešective stress reduction ratio for Toyoura sand under uni-directional cyclic shearing (u=09). The plot shows a wave-shape for each cyclic strain. In this ˆgure, two lines are also shown. The upper line shows G=1.0z and the lower line shows G=0z. During the cyclic shear, the observed plots come in contact with or come close to both lines. These lines show the maximum or the minimum possibility of changes in ešective stress during cyclic shear. Based on this ˆgure, it is also reasonable to conclude that the ešective stress reduction ratio is a function of resultant shear strain G. Further discussion on this phenomenon will be reviewed later. Figure 20 shows the relationship between the cumulative shear strain and the ešective stress reduction ratio for Toyoura sand under multi-directional cyclic shearing (u =909) with dišerent levels of shear strain amplitude. The results show that the larger the shear strain amplitude, the faster the ešective stress decreases. It is interesting to note that in the case of uni-directional shear (as shown in ESTIMATION OF CHANGES IN EFFECTIVE STRESS FOR SATURATED GRANULAR SOIL In the previous chapter, the characteristics of the changes in ešective stress and the post-earthquake settlement in saturated granular materials under multi-directional cyclic shear were discussed. In this chapter, in order to develop an estimation method for the changes in ešective stress and the post-earthquake settlement in saturated granular materials for practical implications, a simple method is proposed. This method consists of a set of equations that can be used to estimate the changes in ešective stress in saturated granular materials as a function of the cumulative shear strain. Equations Showing Changes in EŠective Stress during Cyclic Shear Ohara and Matsuda (1988) proposed a relationship between the excess pore water pressure and the number of cycles in order to estimate the settlement of the saturated clay layer induced by cyclic shear. Further, Matsuda (1997) proposed a simple method for predicting the accumulation of excess pore water in the clay layer during an earthquake. Later, it was found that the relationship between the decrease in ešective stress and the shear strain amplitude subjected to uni-directional cyclic shear under undrained conditions can be formulated by a hyperbola (Matsuda and Nagira, 2000). Based on these reviews, for saturated granular materials subjected to multi-directional shear, the relationship between the

882 MATSUDA ET AL. Table 3. Coe cient of A, B, C, m for Toyoura sand Toyoura sand A B C m D r=70z 1.2 0.02 1.05-0.55 D r=90z 1.8 0.001 1.03-0.7 Table 4. Coe cient of A, B, C, m for GBFS GBFS A B C m D r=70z 1.2 0.02 1.05-0.55 ešective stress reduction ratio and the cumulative shear strain was derived as the following equation: `Ds?v v0 `= G * (3) a+b G* where a and b canbedeˆnedas a=a g m (4) b= g B+C g (5) Parameters A, B, C and m in Eqs. (4) and (5) can be determined empirically by the curve-ˆtting method. The results for Toyoura sand and GBFS are shown in Tables 3 and 4, respectively. Considering the ešect of the phase dišerence, it has been concluded that when shear strain amplitude is relatively high, the ešective stress reduction ratio is not in uenced by the variation in phase dišerence. The higher the shear strain amplitude, the smaller the ešect of the phase dišerence on the ešective stress reduction ratio. Figures 10 and 11 indicate that the application of Eq. (3) is assumed to be more promising for the shear strain amplitude, at least for levels of amplitude higher than g=0.3z. These ˆndings suggest that in order to simulate lower levels of shear strain amplitude, an attempt to derive a modiˆed equation should be made. Estimation of Changes in EŠective Stress under Multidirectional Cyclic Shearing (u=909) Figures 21(a) and (b) show a comparison between the experimental and the calculated results for the ešective stress reduction ratio `Ds? v/s? vo` on Toyoura sand. These results were obtained under a shearing of u=909with dišerent levels of shear strain amplitude. In this study, the specimens with relative densities of D r =70z (referred to as medium-density specimens) and 90z (referred to as high-density specimens) were prepared to investigate the ešect of densities on the cyclic properties of granular materials. It is seen from these ˆgures that the experimental results almost correspond to the calculated ones obtained using Eqs. (3) to (5). By comparing the test results under dišerent relative densities, however, it is apparent that for dense specimens (D r =90z), the ešective stress decreases at a slower rate than that for the specimens with a lower density. This fact suggests that such dense specimens have a higher shear strength and, as a result, an increasing liquefaction resistance. Fig. 21. Comparison between the experimental and calculated values for ešective stress reduction ratio on Toyoura sand with dišerent relative densities under multi-directional cyclic shear (u=909) Fig. 22. Comparison between the experimental and calculated values for ešective stress reduction ratio on GBFS under multi-directional cyclic shear (u=909) Figure 22 shows a comparison between the experimental and the calculated results for the ešective stress reduction ratio for GBFS with D r =70z. The agreements between them are reasonable. This suggests that ešective stress reduction ratios, estimated in such a way, are applicable, at least for granular materials under a multi-

MULTI-DIRECTIONAL CYCLIC SIMPLE SHEAR 883 directional cyclic shear of u=909. Estimation of Changes in EŠective Stress under Unidirectional Cyclic Shearing (u=09) It has been shown in Fig. 19 that the relationship between the ešective stress reduction ratio and the cumulative shear strain under uni-directional cyclic shear (u=09) formed a wave-shaped curve (ruœing curve). Moreover, as discussed in the previous sections, it is possible to develop a relationship between the ešective stress reduction ratio and resultant shear strain G. Based on this standpoint, it is possible to develop an equation for these ruœing components by modifying the original Eq. (3) as follows: `Ds?v v0 `= G* a+b G* -DG h (6) The term DG h refers to the ruœing-curved component, where D is a constant that is proportional with the shape and the size of the ruœing-curved component. Resultant shear strain G refers to the radial distance from the origin to the current position of the specimen and represents the shear strain path on the horizontal plane. When ruœing component DG h can be deˆned as the dišerence between the estimation curve produced from Eq. (3) and the experimental resultant curve, then DG h canbesolvedwith the following equation: log DG h =log D+h log G (7) Figure 23 shows the relationship between the calculated log DG h in Eq. (7) and log G for the GBFS samples under g=1.0z. The log DG h versus log G plot shows a straight line with a certain slope h, andthe``y-intercept'' value corresponds to the constant of D. The two straight lines in Fig. 23 were obtained using the experimental data from the fourth-wave and the thirteenth-wave taken from Fig. 24. This can be done by assuming that the continuous wave-curve, shown in Fig. 24, was developed from a group of several single waves. In Fig. 23, the slope of these two straight lines can be determined as approximately h=2. In addition, since the constant of D is proportional to the number of cycles, the size of the ruœing curves tends to decrease with an increase in G*, and ˆnally becomes asymptotic with the x-axis. Thus, the value of D should be obtained from the condition D=D 0 -E G* (8) This equation satisˆes DÆ0, D 0 is the initial value of D,andE is a constant that is proportional with the rate of the size-decreasing of the waving curve when cumulative shear strain G* increases. To show the practical application of Eqs. (6) to (8), a comparison of the calculated and the experimental results for the ešective stress reduction, related to the cumulative shear strain for GBFS, is shown in Fig. 24. Here, GBFS was chosen because this material exhibits a more uctuating waving curve than that of Toyoura sand. In Fig. 24, D 0 and E are assumed to be 0.09 and 0.00025, respectively. According to this ˆgure, Fig. 23. Relationship between log DG h and log G Fig. 24. Comparison between the experimental and calculated values for ešective stress reduction ratio on GBFS under uni-directional cyclic shear the curve developed from the calculated data agrees well with the experimental results. Thus, the validity of these proposed equations can be conˆrmed. CONCLUSIONS In order to clarify the ešect of the cyclic shear direction on the properties of saturated granular materials, such as the changes in ešective stress and the post-earthquake settlement, several series of cyclic simple shear tests were carried out using a multi-directional cyclic simple shear test apparatus. The main conclusions are summarized as follows. 1) The shear strain amplitude has a signiˆcant ešect on the changes in the ešective stress of granular materials. However, at higher levels of shear strain amplitude, the cyclic shear direction has little in uence on the decrease in ešective stress. 2) The in uence of the phase dišerence on the vertical strain of saturated granular materials after shearing is small under the same cumulative shear strain, regardless of the shear strain amplitude. 3) Higher levels of shear strain amplitude show higher amounts of vertical strain after shearing under the

884 MATSUDA ET AL. same cumulative shear strain. Furthermore, it is found that the vertical strain on the GBFS after shearing was lower than that of the Toyoura sand. 4) In order to evaluate the changes in the ešective stress of saturated granular materials during cyclic shearing, an estimation method under uni-directional and multidirectional cyclic simple shear conditions was proposed. The changes in the ešective stress of saturated granular materials can be predicted with this proposed model by a function of the cumulative shear strain and the resultant shear strain. ACKNOWLEDGEMENTS In this study, some of the experimental works were performed by the graduate students of Yamaguchi University. The authors wish to thank the students for their support. The second author would also like to express his gratitude to the Government of the Republic of Indonesia for ˆnancially supporting his doctoral research. REFERENCES 1) Dobry, R., Ladd, R. S., Yokel, R. Y. and Chung, R. M. 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